straight line. Now the arches of Westminster Bridge are cycloidal, and can therefore be regarded as due to the rolling of a certain circle on a certain straight line. Now suppose we were asked whether this circle actually exists or is a mere fiction. In one sense I answer that it does not exist. So far as I know, no physical circle actually rolled at some date in the world's history on a physical straight-edge to produce the arches of Westminster Bridge. On the other hand, the circle is not a mere fiction. The cycloidal arches really do exist, and the circle corresponding to them is completely determined by the shape and size of these arches. This connexion is a real fact, absolutely independent of our minds and their operations. I therefore say that the circle exists, in the sense that it is a determinate function of the arches, which exist in the ordinary sense. Points, straight lines, etc., as defined by us, exist in the same sense as the circle determined by the arches of Westminster Bridge; the particular series of volumes which define points exist in the same sense as the arches themselves.

Additional works that may be consulted with profit :

A. N. WHITEHEAD, Principles of Natural Knowledge, Part III. Concept of Nature, Cap. IV.

CHAPTER II

Alice sighed wearily. "I think you might do something better with the time," she said, "than waste it asking riddles with no answers.'

"If you knew Time as well as I do," said the Hatter, "you wouldn't talk about wasting it."

(LEWIS CARROLL, Alice in Wonderland.)

The General Problem of Time and Change

We have now said as much about Space as can be said with profit before its relations to Time and Matter have been dealt with. We have shown at least how the concepts, such as points, lines, planes, etc., which are needed, whatever view we finally take of Space, are connected with the rough, untidy facts that we can perceive. We have not, however, explained why there is supposed to be one single Space in which all the events of nature are located, nor how things. have places assigned to them in it. This can only be done at a later stage. In the meanwhile I propose to treat the concepts of Time and Change, as they appear at the same level of thought.

At first sight the problems of Time look very much like those of Space, except that the single dimension of Time, as compared with the three of Space, seems to promise greater simplicity. We shall point out these analogies at the beginning; but we shall find that they are somewhat superficial, and that Time and Change are extremely difficult subjects, in which spatial analogies help us but little.

The physicist conceives Time in much the same way as he conceives Space. Just as he distinguishes Space

from the matter in it, so he distinguishes Time from events. Again, mere difference of position in Time is supposed to have no physical consequences. It is true that, if I go out without my overcoat at 2 A.M., I shall probably catch cold; whilst, if I do so at 2 P.M., I shall probably take no harm. But this difference is never ascribed to the mere difference in date, but to the fact that different conditions of temperature and dampness will be contemporary with my two expeditions. Again, Time, like Space, is supposed to be continuous, and physicists suppose (or did so until quite lately) that there is a single time-series in which all the events of nature take place. This series is of one dimension, so that, as far as appears at present, Time is like a very simple Space consisting of a single straight line.

Just as we treat our geometry in terms of unextended points and their relations, so we treat our chronometry in terms of moments without duration and their relations. Duration in Time corresponds to extension in Space. Now, just as we never perceive points or even unextended particles, so we are never aware of moments or of momentary events. What we are aware of is finite events of various durations. By an event I am going to mean anything that endures at all, no matter how long it lasts or whether it be qualitatively alike or qualitatively different at adjacent stages in its history. This is contrary to common usage, but common usage has nothing to recommend it in this matter. We usually call a flash of lightning or a motor accident an event, and refuse to apply this name to the history of the cliffs at Dover. Now the only relevant difference between the flash and the cliffs is that the former lasts for a short time and the latter for a long time. And the only relevant difference between the accident and the cliffs is that, if successive slices, each of one second long, be cut in the histories of both, the contents of a pair of adjacent slices may be very different in the first case and will be very similar in the second case. Such

merely quantitative differences as these give no good ground for calling one bit of history an event and refusing to call another bit of history by the same name.

Now the temporal relations which we perceive among events are similar to the relations of partial or complete overlapping which we can perceive in the case of two extended objects, like a pair of sticks. The possible time-relations between two events can be completely represented by taking a single straight line, letting "left-to-right" on this stand for "earlier and later," and taking two stretches on this line to represent a pair of finite events. Let AB and CD be two events of which the latter lasts the longer; then the possible temporal relations between the two are represented by the nine figures given below.

The most general kinds of relation between finite events are those of partial precedence and partial subsequence; the rest can be defined in terms of these. From these crude perceptible data and their crude perceptible relations the concepts of momentary events and moments can be obtained, and their exact relations determined, by the Method of Extensive Abstraction. I believe that, as a matter of history, one of the first successful applications of the method was made by Dr Norbert Wiener to this very problem.

The motives that lead us to apply Extensive Abstraction to Time are the same as those which lead us to apply it to Space. As scientists our main interest is to discover laws connecting events of one kind with events of other kinds at different times. Now, just

as the geometrical relations of finite volumes, as such, are of unmanageable complexity, so are the causal relations of events of finite duration. There is no simple relation between the contents of one hour and the contents of another. But the shorter we make our events the simpler become the relations between them. So, finally, we state our laws in terms of socalled "momentary events" and their exact relations, and we "analyse" finite events into sets of momentary ones, and explain their relations in terms of those of their momentary "parts." Everything that has been said of this procedure in geometry applies, mutatis mutandis, to its use in physics. Momentary "events are not really events, any more than points are little volumes. A momentary event is not "part of" a finite one in the plain straightforward sense in which the event of a minute is part of the event that occupies a certain hour. The meanings of all these concepts, and their relations, have to be given in terms of perceptible entities and their relations, by means of Extensive Abstraction.

What we have been saying is most excellently illustrated by the science of Mechanics. What we want to deal with there is the movements of finite bodies, like wheels and planets; and we want to treat their changes of position and motion over long periods of time. To do this we have first to analyse the finite bodies into unextended particles, and then to analyse the finite events into momentary ones. The laws of Mechanics are only simple when they state relations between momentary configurations of one set of particles and a later or earlier configuration of the same or another set of particles. The gap between the perceptible facts, that we are trying to describe and predict, and the imperceptible concepts and relations, in terms of which we have to treat the facts, is bridged by Extensive Abstraction, applied both to extension in Space and to duration in Time. Mechanics is a